bbs10_ppt_ch07.ppt

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 7-1

Chapter 7

Sampling Distributions

Basic Business Statistics10th Edition

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-2

Learning Objectives

In this chapter, you learn: The concept of the sampling distribution

To compute probabilities related to the sample mean and the sample proportion

The importance of the Central Limit Theorem

To distinguish between different survey sampling methods

To evaluate survey worthiness and survey errors




Sampling Distribution of

the Mean

Sampling Distribution of the Proportion



A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population


Developing a Sampling Distribution

Assume there is a population …

Population size N=4

Random variable, X,

is age of individuals

Values of X: 18, 20,

22, 24 (years)

A B C D


.3

.2

.1

0 18 20 22 24

A B C D

Uniform Distribution

P(x)

x

(continued)

Summary Measures for the Population Distribution:


214

24222018

N

Xμ i

2.236N

μ)(Xσ

2i


16 possible samples (sampling with replacement)

Now consider all possible samples of size n=2

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

(continued)


16 Sample Means

1st

Obs2nd Observation

18 20 22 24

18 18,18 18,20 18,22 18,24

20 20,18 20,20 20,22 20,24

22 22,18 22,20 22,22 22,24

24 24,18 24,20 24,22 24,24


1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Sampling Distribution of All Sample Means

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X

Sample Means Distribution

16 Sample Means

_


(continued)

(no longer uniform)

_


Summary Measures of this Sampling Distribution:

Developing aSampling Distribution

(continued)

2116

24211918

N

Xμ i

X

1.5816

21)-(2421)-(1921)-(18

N

)μX(σ

222

2Xi

X


Comparing the Population with its Sampling Distribution

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X 18 20 22 24

A B C D

0

.1

.2

.3

PopulationN = 4

P(X)

X _

1.58σ 21μXX

2.236σ 21μ

Sample Means Distributionn = 2

_


Sampling Distribution of the Mean



the Mean



Standard Error of the Mean

Different samples of the same size from the same population will yield different sample means

A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

Note that the standard error of the mean decreases as the sample size increases

n

σσ

X


If the Population is Normal

If a population is normal with mean μ and

standard deviation σ, the sampling distribution

of is also normally distributed with

and

X

μμX

n

σσ

X


Z-value for Sampling Distributionof the Mean

Z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

Xμσ

n

σμ)X(

σ

)μX(Z

X

X

X


Normal Population Distribution

Normal Sampling Distribution (has the same mean)

Sampling Distribution Properties

(i.e. is unbiased )xx

x

μμx

μ

xμ


Sampling Distribution Properties

As n increases,

decreasesLarger sample size

Smaller sample size

x

(continued)

xσ

μ


If the Population is not Normal

We can apply the Central Limit Theorem:

Even if the population is not normal, …sample means from the population will be

approximately normal as long as the sample size is large enough.

Properties of the sampling distribution:

andμμx n

σσx


n↑

Central Limit Theorem

As the sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

x


Population Distribution

Sampling Distribution (becomes normal as n increases)

Central Tendency

Variation

x

x

Larger sample size

Smaller sample size

If the Population is not Normal(continued)

Sampling distribution properties:

μμx

n

σσx

xμ

μ


How Large is Large Enough?

For most distributions, n > 30 will give a sampling distribution that is nearly normal

For fairly symmetric distributions, n > 15

For normal population distributions, the sampling distribution of the mean is always normally distributed


Example

Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.

What is the probability that the sample mean is between 7.8 and 8.2?


Example

Solution:

Even if the population is not normally distributed, the central limit theorem can be used (n > 30)

… so the sampling distribution of is approximately normal

… with mean = 8

…and standard deviation

(continued)

x

xμ

0.536

3

n

σσx


Example

Solution (continued):(continued)

0.31080.4)ZP(-0.4

363

8-8.2

nσ

μ- X

363

8-7.8P 8.2) X P(7.8

Z7.8 8.2 -0.4 0.4

Sampling Distribution

Standard Normal Distribution .1554

+.1554

Population Distribution

??

??

?????

??? Sample Standardize

8μ 8μX

0μz xX





the Mean



Population Proportions

π = the proportion of the population having some characteristic

Sample proportion ( p ) provides an estimate of π:

0 ≤ p ≤ 1

p has a binomial distribution

(assuming sampling with replacement from a finite population or without replacement from an infinite population)

size sample

interest ofstic characteri the having sample the in itemsofnumber

n

Xp


Sampling Distribution of p

Approximated by a

normal distribution if:

where

and

(where π = population proportion)

Sampling DistributionP( ps)

.3

.2

.1 0

0 . 2 .4 .6 8 1 p

πpμn

)(1σp

ππ

5p)n(1

5np

and


Z-Value for Proportions

n)(1

p

σ

pZ

p

Standardize p to a Z value with the formula:


Example

If the true proportion of voters who support

Proposition A is π = 0.4, what is the probability

that a sample of size 200 yields a sample

proportion between 0.40 and 0.45?

i.e.: if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?


Example

if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

0.03464200

0.4)0.4(1

n

)(1σp

1.44)ZP(0

0.03464

0.400.45Z

0.03464

0.400.40P0.45)pP(0.40

Find :

Convert to standard normal:

pσ


Example

Z0.45 1.44

0.4251

Standardize

Sampling DistributionStandardized

Normal Distribution

if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

Use standard normal table: P(0 ≤ Z ≤ 1.44) = 0.4251

0.40 0p


Reasons for Drawing a Sample

Less time consuming than a census Less costly to administer than a census Less cumbersome and more practical to

administer than a census of the targeted population


Nonprobability Sample Items included are chosen without regard to

their probability of occurrence

Probability Sample Items in the sample are chosen on the basis

of known probabilities

Types of Samples Used


Types of Samples Used

Quota

Samples

Non-Probability Samples

Judgement Chunk

Probability Samples

Simple Random

Systematic

Stratified

ClusterConvenience

(continued)


Probability Sampling

Items in the sample are chosen based on known probabilities

Probability Samples

Simple Random

Systematic Stratified Cluster


Simple Random Samples

Every individual or item from the frame has an equal chance of being selected

Selection may be with replacement or without replacement

Samples obtained from table of random numbers or computer random number generators


Decide on sample size: n Divide frame of N individuals into groups of k

individuals: k=N/n Randomly select one individual from the 1st

group Select every kth individual thereafter

Systematic Samples

N = 64

n = 8

k = 8

First Group


Stratified Samples

Divide population into two or more subgroups (called

strata) according to some common characteristic

A simple random sample is selected from each subgroup,

with sample sizes proportional to strata sizes

Samples from subgroups are combined into one

Population

Divided

into 4

strata

Sample


Cluster Samples

Population is divided into several “clusters,” each representative of the population

A simple random sample of clusters is selected All items in the selected clusters can be used, or items can be

chosen from a cluster using another probability sampling technique

Population divided into 16 clusters. Randomly selected

clusters for sample


Advantages and Disadvantages

Simple random sample and systematic sample Simple to use May not be a good representation of the population’s

underlying characteristics Stratified sample

Ensures representation of individuals across the entire population

Cluster sample More cost effective Less efficient (need larger sample to acquire the

same level of precision)


Evaluating Survey Worthiness

What is the purpose of the survey? Is the survey based on a probability sample? Coverage error – appropriate frame? Nonresponse error – follow up Measurement error – good questions elicit good

responses Sampling error – always exists


Types of Survey Errors

Coverage error or selection bias Exists if some groups are excluded from the frame and

have no chance of being selected

Nonresponse error or bias People who do not respond may be different from those

who do respond

Sampling error Variation from sample to sample will always exist

Measurement error Due to weaknesses in question design, respondent

error, and interviewer’s effects on the respondent


Types of Survey Errors

Coverage error

Non response error

Sampling error

Measurement error

Excluded from frame

Follow up on nonresponses

Random differences from sample to sample

Bad or leading question

(continued)


Chapter Summary

Introduced sampling distributions Described the sampling distribution of the mean

For normal populations Using the Central Limit Theorem

Described the sampling distribution of a proportion Calculated probabilities using sampling distributions Described different types of samples and sampling

techniques Examined survey worthiness and types of survey

errors

bbs10_ppt_ch07.ppt

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